# Eulers Relationship

1. Apr 16, 2010

### geo_alchemist

May be it's a stupid question but I can't figure it out.

according to Eulers Relationship:
ej$$\alpha$$=cos$$\alpha$$+jsin$$\alpha$$

on the other hand I have equation:
Vt=V0cos wt
and it can be rewritten as:
Vt=V0ejwt

where V is voltage in AC (see link below)
http://hyperphysics.phy-astr.gsu.edu/Hbase/electric/impcom.html" [Broken]

In this case cos wt is at the place of cos$$\alpha$$, but what I can't understand is, where did jsin $$\alpha$$ go?

Last edited by a moderator: May 4, 2017
2. Apr 16, 2010

### Hurkyl

Staff Emeritus

3. Apr 16, 2010

### geo_alchemist

that is what I am asking about. and what happens with the imaginary part? And if we can simply neglect it, then why?

4. Apr 16, 2010

### malicx

5. Apr 16, 2010

### Hurkyl

Staff Emeritus
Two miniature ideas to pay attention to:

1. Re(eix) is often more convenient than cos(x)
2. Im(eix) may tell you something else that's interesting

6. Apr 17, 2010

### geo_alchemist

After searching through the web, only thing I could conclude and understand is that after several transformation from ejx=cosx+jsinx I come to:
cosx=(eix+e-ix)/2
but, what I can't understand is the equation:
(eix+e-ix)/2=R{eix}
well, the wikipedia says that:
see: http://en.wikipedia.org/wiki/Electrical_impedance" [Broken] Validity of comples representation.

May be it is simple math, but I can't understand if there is any special meaning of curly brackets.

Last edited by a moderator: May 4, 2017
7. Apr 17, 2010

### HallsofIvy

Staff Emeritus
If z= x+ iy then the real part of z is Rz= x.

It's exactly the same thing as if I say "the x-coordinate of the point (2, 3) is 2".

That's all it is- it's not the { } that is important but the "R".

If $e^{j\omega t}= cos(\omega t)+ j sin(\omega t)$ then the "real part" is $cos(\omega t)$ and the "imaginary part' is $sin(\omega t)$ (notice that both "real part" and "imaginary part" of a complex number are real numbers).

$R(e^{j\omega t}) cos(\omega t)$ and $I(e^{j\omega t}) sin(\omega t)$.